Digital Communications Receiver and Method of Estimating Residual Carrier Frequency Offset In A Received Signal

ABSTRACT

A residual carrier frequency offset of a signal received by a receiver, which has a local frequency oscillator that generates a carrier frequency different from a carrier frequency generated by a local frequency oscillator at a transmitter that transmitted the received signal is estimated using first and second channel impulse responses derived from first and second portions of the received signal, an estimated phase difference between the first and second channel impulse responses, and an estimate of the residual carrier frequency offset computed using the estimated phase differences. The residual carrier frequency offset is estimated after most of the carrier frequency offset has been removed from the received signal using conventional means.

FIELD OF THE INVENTION

This invention relates in general to digital communications and, inparticular, to a digital communications receiver and method of removingresidual carrier frequency offset from a received signal.

BACKGROUND OF THE INVENTION

Digital communications is critical to the success of moderncommunications technology. Orthogonal frequency division multiplexing(OFDM) is a viable modulation scheme for packet communications. It hasbeen adopted in a number of communication standards including DSL,802.11 WiFi, 802.16 WiMAX, 802.22 WRAN, and 3GPP LTE etc. One of themajor limitations of OFDM is frequency synchronization errors caused byfrequency differences between a local oscillator in a transmitter andthe local oscillator in a receiver of an OFDM modem. This frequencydifference is called carrier frequency offset (CFO). CFO contributes toa number of impairments to OFDM system performance, such as intercarrier interference (ICI) among OFDM subcarriers, attenuation andconstellation rotation. Needless to say, one important signal processingfunction in an OFDM receiver is to estimate and remove CFO.

In an OFDM system, a transmitted data sequence usually includes atraining sequence (preamble) and predefined pilot subcarriers in eachOFDM symbol. For example, in the proposed 802.22 standard, a superframepreamble is transmitted at the beginning of each superframe, and a framepreamble is transmitted at the beginning of each frame. Each superframecontains 16 frames and each frame contains 25 to 41 OFDM symbols and hasa duration of 10 ms. Moreover, in each OFDM symbol 1680 subcarriers areused, and 240 of those subcarriers are pilot subcarriers. The valuetransmitted on the pilot subcarriers is predetermined and known to thereceiver.

Classical CFO estimation methods examine information in the OFDMpreambles to detect and estimate CFO. However, channel imperfections andnoise deteriorate CFO estimation accuracy, and residual CFO remains inthe received data sequence after CFO estimation and removal has beenperformed. Since there are usually a large number of OFDM data symbolsbetween two preambles, the residual CFO may result in significantperformance degradation.

The detection of residual CFO in the frequency domain is known, astaught in U.S. Pat. No. 7,526,020 which issued on Apr. 28, 2009 to Kaoet al.

SUMMARY OF THE INVENTION

Embodiments of the invention provide a digital communications receiverand method of estimating residual carrier frequency offset by examiningsignal information in the time domain.

One aspect of the invention therefore provides a method of estimating aresidual carrier frequency offset in a signal received by a receiverwhich has a local frequency oscillator that generates a carrierfrequency different from a carrier frequency generated by a localfrequency oscillator at a transmitter that transmitted the receivedsignal, the method comprising: computing first and second channelimpulse responses using first and second portions of the receivedsignal; estimating a phase difference between the first and secondchannel impulse responses; and computing the estimate of the residualcarrier frequency offset using the computed phase differences.

A further aspect provides a method of estimating a residual carrierfrequency offset in a orthogonal frequency division multiplexing (OFDM)signal received by an OFDM receiver after most of the carrier frequencyoffset has been removed from the received signal, comprising:sequentially receiving a Fast Fourier Transform of a pair of OFDMsymbols embodied in the signal received by the OFDM receiver; for eachof the respective OFDM symbols in the pair, computing a channelfrequency response for pilot subcarriers output by the Fast FourierTransform, and assigning a channel frequency response of zero to datasubcarriers output by the Fast Fourier Transform; performing an InverseFast Fourier Transform on the channel frequency responses for therespective OFDM symbols in the pair to obtain a channel impulse responsefor each OFDM symbol; estimating a phase difference between the channelimpulse response computed for the respective OFDM symbols in the pair;and estimating the residual carrier frequency offset using the estimatedphase difference.

Yet a further aspect provides a digital communications receiver,comprising: coarse carrier frequency offset estimation and removal;residual carrier frequency offset estimation that estimates the residualcarrier frequency offset by computing a channel frequency response forpilot subcarriers in a first and second portion of a digitalcommunications signal received by the receiver, performing an InverseFast Fourier Transform on the respective channel frequency responses toobtain respective first and second channel impulse responses, estimatinga phase difference between the first and second channel impulseresponses, and estimating the residual carrier frequency offset usingthe phase difference; and residual carrier frequency offset removal thatuses the estimated residual carrier frequency offset to remove theresidual carrier frequency offset from the communications signal.

An aspect still further provides an OFDM receiver, comprising: a coarsecarrier frequency offset estimation module; a coarse carrier frequencyoffset removal module; a residual carrier frequency offset estimationmodule that estimates a residual carrier frequency offset after thecoarse carrier frequency offset has been estimated and removed, bycomputing a channel frequency response for pilot subcarriers in a pairof OFDM symbols received by the OFDM receiver, performing an InverseFast Fourier Transform on the respective channel frequency responses toobtain respective first and second channel impulse responses, estimatinga phase difference between the first and second channel impulseresponses, and estimating the residual carrier frequency offset usingthe estimated phase difference.

BRIEF DESCRIPTION OF THE DRAWINGS

Having thus generally described the nature of the invention, referencewill now be made to the accompanying drawings, in which:

FIG. 1 is a schematic diagram of one embodiment of a digitalcommunications receiver provisioned with residual CFO estimation inaccordance with an embodiment of the invention;

FIG. 2 is a flow diagram of a method of residual CFO estimation inaccordance with an embodiment of the invention; and

FIG. 3 is a graph representing simulation results of variousimplementations of the method of residual CFO estimation in accordancewith an embodiment of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In one embodiment, the invention is implemented in a discrete-timebaseband OFDM system, and the invention will be explained below withreference to OFDM. However, it should be understood that the inventionis not limited to OFDM and can be implemented in receivers that aredesigned or provisioned to operate using other communications protocols.In an embodiment of the invention, pilot symbols in OFDM data symbolsare used to estimate a channel impulse response (CIR) of a receivedsignal. Residual CFO is estimated using the estimated CIR derived fromat least one pair of data symbols. Averaging can be used if multiplepairs of data symbols are used to estimate the residual CFO. Averagingfurther improves the residual CFO estimation accuracy. Residual CFOestimation in accordance with embodiments of the invention has theadvantage of high accuracy, does not require the pilot symbols to be onthe same subcarriers in different OFDM symbols, and can be applied toany OFDM system which uses pilot symbol subcarriers interspersed withdata symbol subcarriers.

FIG. 1 is a schematic functional block diagram of one exemplaryembodiment of a receiver 20 provisioned with residual CFO estimation inaccordance with an embodiment of the invention. A transmitter 22transmits a communications signal 24 to the receiver 20. In oneembodiment, the transmitter is an OFDM transmitter that transmits anOFDM signal that is received by an antenna 26 of the receiver 20, andthe receiver 20 is an OFDM receiver. A radio frequency (RF) front end28, well known in the art, includes an RF module which downconverts andshapes the received signal in accordance with the requirements of ananalog-to-digital (A/D) converter 30. The A/D converter 30 samples thesignal at a predetermined rate and outputs a digital signal that ispassed to a digital downcoverter and decimator 32, which converts thedigital signal to baseband in a manner satisfying the Nyquist rule. Thebaseband signal is passed to a symbol timing and synchronization module34 that determines a position and extent of each OFDM symbol in thebaseband signal, also in a manner well known in the art. A coarse CFOestimation module 36 performs coarse CFO estimation using any one ofmany algorithms that are well known in the art. The estimated coarse CFOinformation is passed to a coarse CFO removal module 38 that uses a timedomain multiplier function well known in the art to remove most of theCFO from the baseband signal. In some embodiments the coarse CFOestimation and removal functions are performed by a single module. Thecyclic prefixes are then removed from the baseband signal and thebaseband signal is input in parallel on an OFDM symbol-by-symbol basisto a Fast Fourier Transform (FFT) 42 and an OFDM symbol buffer 50. TheFFT 42 processes the baseband signal in the frequency domain. Output ofthe FFT 42 is passed to a pilot extraction module 44.

The pilot extraction module 44 extracts pilot subcarriers from the FFToutput and computes a channel frequency response (H′(k)) for each pilotsubcarrier, as will be explained below in more detail with reference toFIG. 2. Output from the pilot extraction module 44 is passed to anInverse Fast Fourier Transform (IFFT) 46 which performs an IFFT on(H′(k)) to obtain a time domain channel impulse response (CIR) for theOFDM symbol. The CIR is passed to a residual CFO estimation module 48which estimates the residual CFO using the methods described below withreference to FIGS. 2 and 3.

The symbol buffer module 50, which can be implemented as afirst-in-first-out (FIFO) memory register, for example, stores each OFDMsymbol output by the cyclic prefix removal module 40 for a predeterminedperiod of time that is dependent on the residual CFO estimationimplementation, as will be explained below in more detail with referenceto FIGS. 2 and 3, to permit residual CFO estimation to be performed onsome of the OFDM symbols. Output from the residual CFO estimation module48 is passed to a residual CFO removal module 52 that receives thedelayed symbols from the symbol buffer module 50, and removes residualCFO using a time domain multiplier function that functions in the sameway as the coarse CFO removal function. Output from the residual CFOremoval module 52 is passed to a FFT module 53 that transforms the timedomain signal into the frequency domain. Output of the FFT module 53 ispassed to a channel estimation module 54 which outputs channelestimation information used for signal equalization by a signalequalization module and subsequent signal processing 58. Output of theFFT module 53 is also passed to a data subcarrier extraction module 56.The data subcarrier extraction module 56 extracts the data subcarriersfrom the signal, also in a manner well known in the art. The output fromthe data subcarrier extraction module 56 is passed to the signalequalization module and subsequent signal processing 58, which is wellknown in the art and will not be explained in any further detail. Otherwell known components, modules and functions of the receiver 20 are notspecifically illustrated nor described. Instead, they are generallyrepresented by the receiver 20.

FIG. 2 is a flow diagram of the method of residual CFO estimation inaccordance with an embodiment of the invention. In estimating residualCFO in accordance with embodiments of the invention, it is assumed thatany residual CFO is much smaller than a subcarrier spacing between twoneighboring subcarriers in the received signal. It is also assumed thatsymbol timing synchronization in the receiver is well designed, i.e.,aside from thermal noise the residual CFO is the principal cause ofsystem performance degradation.

As is well known in the art, each OFDM symbol consists of N subcarriers.The OFDM radio channel is a frequency selective multipath fading channelwith a time domain channel impulse response function h(n) with pathN_(p) components such that:

$\begin{matrix}{{{h(n)} = {\sum\limits_{l = {c\; 1}}^{c\; 2}\; {h_{l}{\delta ( {n - n_{1}} )}}}},} & {{eq}.\mspace{11mu} 1}\end{matrix}$

where: h_(l) and n_(l) are respectively the complex-valued gain andrelative delay of the l th path. The Doppler shift in the radio channelis considered a part of the path gain. The path gain is assumed to betime invariant over the OFDM symbols used to compute the residual CFO.It is also assumed that h(n) includes a time offset generated by thesymbol timing synchronization module 34 (FIG. 1). It is further assumedthat the symbol timing synchronization module 34 can estimate the valuesof n_(c1) and n_(c2).

If there is no residual CFO, the received signal on the k^(th)subcarrier can be expressed as

Y(k)=H(k)X(k)+W(k),  eq. 2

where:kε[0,N−1];N is the FFT size used in the OFDM system;H(k) is the channel frequency response;X(k) is the value modulated onto the subcarrier; andW(k) is the additive white Gaussian noise (AWGN) at the k^(th)subcarrier.

The AWGN at different subcarriers is assumed to be independent of therespective subcarriers and equally distributed across the respectivesubcarriers.

When residual CFO exists, the received signal in the time domain can beexpressed as:

$\begin{matrix}\begin{matrix}{{y^{\prime}(n)} = {{\frac{1}{N}\lbrack {\sum\limits_{k = 0}^{N - 1}\; {{H(k)}{X(k)}^{{j2\pi}\; {n{({k + v})}}N}}} \rbrack} + {w(n)}}} \\{{= {{{y(n)}^{{j2\pi}\; {v/N}}} + {w(n)}}},}\end{matrix} & {{eq}.\mspace{11mu} 3}\end{matrix}$

Where:

nε[0,N−1];v is the CFO normalized by the subcarrier spacing;w(n) is the time domain AWGN signal; and

${y(n)} = {\frac{1}{N}\lbrack {\sum\limits_{k = 0}^{N - 1}\; {{H(k)}{X(k)}^{{j2\pi}\; {n/N}}}} \rbrack}$

is the time domain received signal without residual CFO.

The received frequency-domain signal on the k^(th) subcarrier affectedby residual CFO is:

$\begin{matrix}\begin{matrix}{{Y^{\prime}(k)} = {\sum\limits_{l = 0}^{N - 1}{{y^{\prime}(l)}^{{- j}\; 2\pi \; {{kl}/N}}}}} \\{= {\sum\limits_{l = 0}^{N - 1}{{H(l)}{X(l)}\frac{\sin ({\pi v})}{N\; \sin \; ( {{\pi ( {l - k + v} )}/N} )}}}} \\{{{^{j\; \pi \; {{v{({N - 1})}}/N}}^{{- j}\; {{\pi {({j - k})}}/N}}} + {{W(k)}.}}}\end{matrix} & {{eq}.\mspace{14mu} 4}\end{matrix}$

Equation 4 can be rewritten as:

$\begin{matrix}{{Y^{\prime}(k)} = {{{H(k)}{X(k)}\frac{\sin ({\pi v})}{N\; \sin \; ( {{\pi ( {l - k + v} )}/N} )}^{j\; \pi \; {{v{({N - 1})}}/N}}^{{- j}\; {{\pi {({j - k})}}/N}}} + {{{W(k)}.\mspace{79mu} {where}}\text{:}}}} & {{eq}.\mspace{14mu} 5} \\{{ICI}_{k} = {\sum\limits_{\underset{l \neq k}{l = 0}}^{N - 1}{{H(l)}{X(l)}\frac{\sin ({\pi v})}{N\; \sin \; ( {{\pi ( {l - k + v} )}/N} )}^{j\; \pi \; {{v{({N - 1})}}/N}}^{{- j}\; {{\pi {({j - k})}}/N}}}}} & {{eq}.\mspace{14mu} 6}\end{matrix}$

Equation 6 represents the intercarrier interference (ICI) caused by theresidual CFO. Since the residual CFO is small compared to the subcarrierspacing, the ICI is also small. When the AWGN is moderate, the receivedsignal can be approximated as:

$\begin{matrix}{{{Y^{\prime}(k)} \approx {{H(k)}{X(k)}\frac{\sin ({\pi v})}{N\; \sin \; ( {\pi \; {v/N}} )}^{j\; \pi \; {{v{({N - 1})}}/N}}}},} & {{eq}.\mspace{14mu} 7}\end{matrix}$

As shown in FIG. 2, when the pilot extraction module 44 receives theoutput from the FFT 42, it determines at 62 if whether the respectivesubcarriers in the received signal are pilot subcarriers or datasubcarriers using predetermined information based on the OFDM protocolof the transmitted signal. If it is determined that a particularsubcarrier is not a pilot subcarrier, the variable (H′(k)) associatedwith that subcarrier is set equal to zero at 64. If, however, it isdetermined at 62 that the subcarrier is a pilot subcarrier, the receivedsubcarrier signal Y′(k) is divided by the known value X(k) transmittedon the pilot subcarrier to compute the channel frequency response(H′(k)), as follows:

$\begin{matrix}{{{H^{\prime}(k)} = {\frac{Y^{\prime}(k)}{X(k)} \approx {{H(k)}\frac{\sin ({\pi v})}{N\; \sin \; ( {\pi \; {v/N}} )}^{j\; \pi \; {{v{({N - 1})}}/N}}}}},} & {{eq}.\mspace{14mu} 8}\end{matrix}$

It is then determined at 68 whether the last subcarrier in the OFDMsymbol has been examined. If not, the process returns to 60 and the nextsubcarrier is examined as explained above. If so, an IFFT is performedat 70 on H′(k) to obtain a time domain channel impulse response (CIR)h′(n). It is assumed that the pilot index pattern is designed in such away that in the time domain h′(n)≈h_(all)′(n) when nε[n_(c1),n_(c2)],i.e. the residual CFO associated with the pilot subcarriers isrepresentative of the CFO associated with the data subcarriers.

It is then determined at 72 whether a last OFDM symbol in aimplementation-specific sequence of consecutive OFDM symbols used toestimate the residual CFO has been examined. As will be explained belowin more detail, residual CFO is estimated using the time domain CIR ofat least one pair of OFDM symbols selected from a group of consecutiveOFDM symbols.

For example, a 2^(nd) OFDM symbol closely following the above-referencedOFDM symbol is examined. The start index of the second OFDM symbol has atime offset of n₁ samples compared to the start index of the firstsymbol examined. It is assumed that the channel remains constant for thefirst and second OFDM symbols. The time domain received signal for thesecond symbol can be expressed as

$\begin{matrix}\begin{matrix}{{y_{1}^{''}( {n + n_{1}} )} = {{{\frac{1}{N}\lbrack {\sum\limits_{k = 0}^{N - 1}{{H(k)}{X_{1}(k)}^{j\; 2\pi \; {{nk}/N}}}} \rbrack}^{j\; 2\pi \; {{nv}/N}}^{{j2}\; n_{1}{v/N}}} +}} \\{{w( {n + n_{1}} )}} \\{= {{{y_{1}(n)}^{j\; 2\pi \; {{nv}/N}}^{{j2}\; \pi \; n_{1}{v/N}}} + {w( {n + n_{1}} )}}} \\{{= {{{y_{1}^{\prime}(n)}^{{j2\pi}\; n_{1}{v/N}}} + {w( {n + n_{1}} )}}},}\end{matrix} & {{eq}.\mspace{14mu} 10}\end{matrix}$

Where:

nε[0,N−1];X₁(k) is the value modulated on to the k th subcarrier;

${y_{1}(n)} = {\frac{1}{N}\lbrack {\sum\limits_{k = 0}^{N - 1}{{H(k)}{X_{1}(k)}^{j\; 2\pi \; {{nk}/N}}}} \rbrack}$

is the received signal without CFO; and y′₁(n)=y₁(n)e^(j2πnv/N) is thereceived signal with CFO if the 2nd symbol is not time shifted by n₁samples.

For the second symbol, the received frequency-domain signal on thek^(th) subcarrier affected by CFO is:

$\begin{matrix}{{Y_{1}^{\prime}(k)} = {\sum\limits_{l = 0}^{N - 1}{{y_{1}^{''}( {l + n_{1}} )}{^{{- {j2\pi}}\; {{kl}/N}}.}}}} & {{eq}.\mspace{14mu} 11}\end{matrix}$

Using the same procedure used to obtain H′(k), Y″(k) is divided by X₁(k)on the pilot subcarriers to obtain:

$\begin{matrix}{{{H_{1}^{''}(k)} = {\frac{Y_{1}^{''}(k)}{X_{1}(k)} \approx {{H^{\prime}(k)}^{j\; 2\pi \; n_{1}{v/N}}}}},} & {{eq}.\mspace{14mu} 12}\end{matrix}$

As explained above, for each data subcarrier in the second OFDM symbol,H″(k) is set to zero at 64. An IFFT is performed on H″₁(k). at 70 toobtain the time domain CIR h″₁(n). For the same time rangenε[n_(c1),n_(c2)], it should be noted that h″₁(n)n)≈h′(n)e^(j2πn) ¹^(v/N).

A phase difference between h′(n) and h″₁(n) is estimated at 74 asfollows:

φ is the phase difference between h′(n) and h″₁(n).φ is estimated as follows:

$\begin{matrix}{\varphi = {- {{\underset{n \in {\lbrack{n_{c\; 1},n_{c\; 2}}\rbrack}}{mean}( {{h^{\prime}(n)}*{{conjugate}( {h_{1}^{''}(n)} )}} )}.}}} & {{eq}.\mspace{14mu} 12}\end{matrix}$

The residual CFO value v′ is then estimated at 76 using the followingformula:

$\begin{matrix}{v^{\prime} = {\frac{1}{2\pi}\frac{N}{n_{1}}\varphi}} & {{eq}.\mspace{14mu} 13}\end{matrix}$

Note that φ usually has a detection range of φε(−π,+π). The time offsetbetween the two symbols is an integer number of OFDM symbols, thereforethe residual CFO detection range can be expressed as:

$v^{\prime} \in {( {{- \frac{N}{2n^{\prime}}},{+ \frac{N}{2n^{\prime}}}} )\mspace{14mu} \ldots}$

As explained above, residual CFO estimation uses a pair of OFDM symbolshaving a variable separation between them. If the receiver 20 isprovisioned with logic for computing the residual carrier frequencyoffset as described above using multiple pairs of consecutive OFDMsymbols, the estimation accuracy can be improved by averaging estimationresults for two or more of the pairs of OFDM symbols. For example, witha sequence of OFDM symbols labeled Symbol 1, 2, 3, 4, 5 and 6, thereceiver 20 can be programmed to perform residual carrier frequencyoffset estimation using symbol pairs (1, 4), (2, 5) and (3, 6). Thethree estimated residual CFO values are then averaged to yield anestimate of the residual CFO that is generally more accurate than anestimate derived from a single OFDM symbol pair. As was discussed abovein connection with FIG. 1, the residual CFO is then removed using theestimate of he residual CFO.

FIG. 3 is a graph representing the simulated results of variousimplementations using different numbers of pairs of OFDM symbols toestimate residual CFO in accordance with an embodiment of the invention.The results show that increasing the number of OFDM symbol pairs used toestimate residual CFO improves the residual CFO estimation accuracy, butdecreases a range within which residual CFO can be detected.

Accordingly, the present invention is not limited to only thoseimplementations described above. Those of skill in the art willappreciate that the various illustrative modules, functional blocks andmethod steps described in connection with the above described figuresand the implementations disclosed herein can often be implemented aselectronic hardware, software, firmware or combinations of theforegoing. To clearly illustrate this interchangeability of hardware andsoftware, various illustrative modules and method steps have beendescribed above generally in terms of their functionality. Whether suchfunctionality is implemented as hardware or software depends upon theparticular application and design constraints imposed on the overallsystem. Skilled persons can implement the described functionality invarying ways for each particular application, but such implementationdecisions should not be interpreted as causing a departure from thescope of the invention. In addition, the grouping of functions within amodule, functional block or step is for ease of description. Specificfunctions can be moved from one module, functional block or step toanother without departing from the invention.

The various illustrative modules, functional blocks and method stepsdescribed in connection with the implementations disclosed herein can beimplemented or performed with a processor, a digital signal processor(“DSP”), an application specific integrated circuit (“ASIC”), a fieldprogrammable gate array (“FPGA”) or other programmable logic device,discrete gate or transistor logic, discrete hardware components, or anycombination thereof designed to perform the functions described herein.A processor can be a microprocessor, but in the alternative, theprocessor can be any processor, controller, or microcontroller. Aprocessor can also be implemented as a combination of computing devices,for example, a combination of a DSP and a microprocessor, a plurality ofmicroprocessors, one or more microprocessors in conjunction with a DSPcore, or any other such configuration.

Additionally, the steps of a method or algorithm described in connectionwith the implementations disclosed herein can be embodied directly inhardware, in a software module executed by a processor, or in acombination of the two. A software module can reside in computer ormachine readable storage media such as RAM memory, flash memory, ROMmemory, EPROM memory, EEPROM memory, registers, hard disk, a removabledisk, a CD-ROM, or any other form of computer readable storage mediumincluding a network storage medium. An exemplary storage medium can becoupled to the processor such that the processor can read informationfrom, and write information to, the storage medium. In the alternative,the storage medium can be integral to the processor. The processor andthe storage medium can also reside in an ASIC.

The above description of the disclosed implementations is provided toenable any person skilled in the art to make or use the invention.Various modifications to these implementations will be readily apparentto those skilled in the art, and the generic principles described hereincan be applied to other implementations without departing from thespirit or scope of the invention. Thus, it is to be understood that thedescription and drawings presented herein represent exampleimplementations of the invention and are therefore representative of thesubject matter which is broadly contemplated by the present invention.It is further understood that the scope of the present invention fullyencompasses other implementations.

1. A method of operating a receiver for estimating a residual carrierfrequency offset in a signal received by the receiver which has a localfrequency oscillator that generates a carrier frequency different from acarrier frequency generated by a local frequency oscillator at atransmitter that transmitted the received signal, the method comprising:computing first and second channel impulse responses using first andsecond portions of the received signal; estimating a phase differencebetween the first and second channel impulse responses; and computingthe estimate of the residual carrier frequency offset using the computedphase differences.
 2. The method as claimed in claim 1, wherein thereceived signal is an orthogonal frequency division multiplexing (OFDM)signal containing OFDM symbols.
 3. The method as claimed in claim 2,wherein the first and second channel impulse responses are computedusing respective pilot subcarriers in the OFDM symbols.
 4. The method asclaimed in claim 3, wherein the channel impulse responses on pilotsubcarriers are computed by dividing the received signal by a knowntransmitted signal on the respective pilot subcarriers.
 5. The method asclaimed in claim 2, further comprising averaging the estimated residualcarrier frequency offsets computed with respect to multiple pairs ofOFDM symbols to improve an accuracy of the residual CFO estimation. 6.The method as claimed in claim 1 wherein estimating the phase differencebetween the first and second channel impulse responses comprisescomputing:${\varphi = {- {\underset{n \in {\lbrack{n_{c\; 1},n_{c\; 2}}\rbrack}}{mean}( {{h^{\prime}(n)}*{{conjugate}( {h_{1}^{''}(n)} )}} )}}};$=m^(ean)i(h¹(n)* conjugate(hc(n))); n E[li_(c1),n_(c2) where: h′(n) isthe first channel impulse response; h″₁(n) is the second channel impulseresponse; and φ is the phase difference between h′(n) and h″₁(n).
 7. Themethod as claimed in claim 1 wherein computing the estimate of theresidual carrier frequency offset using the estimated phase differencescomprises computing:${v^{\prime} = {\frac{1}{2\pi}\frac{N}{n_{1}}\varphi}},$ where: v′ isthe estimated residual CFO; N is an FFT size used in the OFDM system;and n₁ is the time offset in FFT samples between a start index of thefirst portion of the received signal and a start index of the secondportion of the received signal.
 8. A method of operating an orthogonalfrequency division multiplexing (OFDM) receiver for estimating aresidual carrier frequency offset in an OFDM signal received by the OFDMreceiver after most of the carrier frequency offset has been removedfrom the received signal, comprising: sequentially receiving a FastFourier Transform of a pair of OFDM symbols embodied in the signalreceived by the OFDM receiver; for each of the respective OFDM symbolsin the pair, computing a channel frequency response for pilotsubcarriers output by the Fast Fourier Transform, and assigning achannel frequency response of zero to data subcarriers output by theFast Fourier Transform; performing an Inverse Fast Fourier Transform onthe channel frequency responses for the respective OFDM symbols in thepair to obtain a channel impulse response for each OFDM symbol;estimating a phase difference between the channel impulse responsecomputed for the respective OFDM symbols in the pair; and estimating theresidual carrier frequency offset using the estimated phase difference.9. The method as claimed in claim 8 wherein the pair of the respectiveOFDM symbols comprises adjacent OFDM symbols in the received OFDMsignal.
 10. The method as claimed in claim 8 wherein the pair of therespective OFDM symbols comprises non-adjacent OFDM symbols in thereceived OFDM signal.
 11. The method as claimed in claim 10 wherein thenon-adjacent symbols in the received OFDM signal comprise OFDM symbolsselected from a group of consecutive OFDM symbols in the receivedsignal.
 12. The method as claimed in claim 8 wherein computing a channelfrequency response for the pilot subcarriers output by the Fast FourierTransform comprises: estimating the received signal using:$\begin{matrix}{{y^{\prime}(n)} = {{\frac{1}{N}\lbrack {\sum\limits_{k = 0}^{N - 1}{{H(k)}{X(k)}^{j\; 2\pi \; {{n{({k + v})}}/N}}}} \rbrack} + {w(n)}}} \\{{= {{{y(n)}^{{j2\pi}\; {{nv}/N}}} + {w(n)}}},}\end{matrix}$ where: nε[0,N−1]; N is the FFT size used in the OFDMsystem; v is the residual CFO normalized by the subcarrier spacing; w(n)is the time domain AWGN; and${y(n)} = {\frac{1}{N}\lbrack {\sum\limits_{k = 0}^{N - 1}{{H(k)}{X(k)}^{j\; 2\pi \; {{nk}/N}}}} \rbrack}$is the time domain received signal without residual CFO; computing thechannel frequency response (H′k) using:${{H^{\prime}(k)} = \frac{Y^{\prime}(k)}{X(k)}},$ where: X(k) is aknown value modulated onto the pilot subcarrier.
 13. The method asclaimed in claim 12 wherein estimating the phase difference between thechannel impulse responses computed for the pair of the respective OFDMsymbols is performed using:${\varphi = {- {\underset{n \in {\lbrack{n_{c\; 1},n_{c\; 2}}\rbrack}}{mean}( {{h^{\prime}(n)}*{{conjugate}( {h_{1}^{''}(n)} )}} )}}};$where: h′ (n) is the channel impulse response for a first OFDM symbol ofthe pair; and h″(n) is the channel impulse response for a second OFDMsymbol of the pair.
 14. The method as claimed in claim 13 whereinestimating the residual carrier frequency offset (v′) using theestimated phase difference, is performed using:${v^{\prime} = {\frac{1}{2\pi}\frac{N}{n_{1}}\varphi}};$ where: N isthe FFT size used in the OFDM system; and n₁ is the time offset in FFTsamples between a start index of the first OFDM symbol and a start indexof the second OFDM symbol.
 15. The method as claimed in claim 11 furthercomprising averaging the computed estimate of the residual carrierfrequency offset computed with respect to multiple pairs of non-adjacentOFDM symbols in the group.
 16. A digital communications receiver,comprising: a coarse carrier frequency offset estimation and removalmodule; a residual carrier frequency offset estimation module thatestimates the residual carrier frequency offset by computing a channelfrequency response for pilot subcarriers in a first and second portionof a digital communications signal received by the receiver, performingan Inverse Fast Fourier Transform on the respective channel frequencyresponses to obtain respective first and second channel impulseresponses, estimating a phase difference between the first and secondchannel impulse responses, and estimating the residual carrier frequencyoffset using the phase difference; and a residual carrier frequencyoffset removal module that uses the estimated residual carrier frequencyoffset to remove the residual carrier frequency offset from thecommunications signal.
 17. The digital communications receiver asclaimed in claim 16 wherein the communications signal comprises an OFDMsignal.
 18. The digital communications receiver as claimed in claim 16further comprising a delay for delaying the first and second portions ofthe received signal while the residual frequency offset estimationestimates the residual frequency offset associated with the first andsecond portions of the received signal.
 19. An OFDM receiver,comprising: a coarse carrier frequency offset estimation module; acoarse carrier frequency offset removal module; a residual carrierfrequency offset estimation module that estimates a residual carrierfrequency offset after the coarse carrier frequency offset has beenestimated and removed, by computing a channel frequency response forpilot subcarriers in a pair of OFDM symbols received by the OFDMreceiver, performing an Inverse Fast Fourier Transform on the respectivechannel frequency responses to obtain respective first and secondchannel impulse responses, estimating a phase difference between thefirst and second channel impulse responses, and estimating the residualcarrier frequency offset using the estimated phase difference.
 20. TheOFDM receiver as claimed in claim 19 further comprising a buffer forstoring the pair of OFDM symbols while the residual frequency offsetestimation estimates the residual frequency offset associated with thepair of OFDM symbols.
 21. The OFDM receiver as claimed in claim 20further comprising a residual carrier frequency offset removal modulethat uses the estimated residual carrier frequency offset to remove theestimated residual carrier frequency offset.
 22. The OFDM receiver asclaimed in claim 19 wherein the residual carrier frequency offsetestimation module further comprises logic for selecting at least twopairs from a plurality of consecutive OFDM symbols in the received OFDMsignal to estimate the residual carrier frequency offset.